We consider a contraction map T of the Meir-Keeler type on the union of p subsets A 1 , . . . , A p , p ≥ 2 , of a metric space X, d to itself. We give sufficient conditions for the existence and convergence of a best proximity point for such a map.
Let (X, d) be a metric space, and A1, A2,. .. , Ap be nonempty subsets of X. We introduce a self map T on X, called p-cyclic orbital contraction map on the union of A1, A2,. .. , Ap, and obtain a unique best proximity point of T , that is, a point x ∈ ∪ p i=1 Ai such that d(x, T x) = dist(Ai, Ai+1), 1 i p, where dist(Ai, Ai+1) = inf{d(x, y): x ∈ Ai, y ∈ Ai+1}.
In this manuscript, we introduce the concept of Ω -class of self mappings on a metric space and a notion of p-cyclic complete metric space for a natural number ( p ≥ 2 ) . We not only give sufficient conditions for the existence of best proximity points for the Ω -class self-mappings that are defined on p-cyclic complete metric space, but also discuss the convergence of best proximity points for those mappings.
Rise of antibiotic resistant pathogenic bacteria namely Methicillin Resistant Staphylococcus aureus (MRSA) and Multiple drug resistant Escherichia coli (MDR E. coli results in reduced efficacy of currently used antibacterial agents. Medicinal plants serve as potential targets for biologically effective antibacterial agents. The present study determined the phytochemical and invitro antibacterial activity of ethanol, chloroform, hexane and water extracts of whole plant of Andrographis paniculata against MRSA and MDR Escherichia coli. Zone of inhibition diameters were measured. Compared to all the extracts, ethanolic extract showed highest activity. The antibacterial activity was absent in hexane and water extracts. Chloroform extracts showed moderately good activity. The antibacterial compounds found in ethanolic extract were flavanoids, saponins and alkaloids.
Let A 1 , A 2 , ..., A p (p ∈ N) be non empty subsets of a metric space (X, d). In this paper, a map T : ∪ p i=1 A i → ∪ p i=1 A i , called p-cyclic orbital contraction of Boyd-Wong type is introduced. Convergence of a unique fixed point and a best proximity point for this map are obtained in a uniformly convex Banach space settings. Moreover, the obtained best proximity point is the unique periodic point of the map.
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